For samples of the values $d \in [0,100]$ and the fixed value $k=1000$, plot the four eigenvalues on the complex plane.¶

Bonus question: there is a bifurcation in the diagram above. Can you find the bifurcation point programmatically?¶

In technical applications there occurs often linear systems of the form
$$
\dot x(t) = A x(t) + B u(t)
$$
where $u$ is an given input signal. $x$ is called the state. From the state some quantities $y(t)$ can be
measured, this is decribed by the equation
$$
y(t)=C x(t).
$$
We assume here that the input signal is an harmonic oscillation $u(t)=\sin(\omega t)$ with a given frequency $\omega$ and amplitude one. Then, $y(t)$ is again a harmonic oscillation with the same frequency, but another amplitude. The amplitude depends on the frequency.

The relationship between the in- and out-amplitude is given by the formula
$$
\mathrm{amplitude}(\omega)=\\|(G(i\omega))\\|\quad\text{where}
\quad G(i\omega)=C \cdot (i\omega I -A)^{-1} \cdot B
$$
and $i$ is the imaginary unit.